Finite element approximation of the Neumann Problem with singular boundary data

We consider in this work the finite element approximation of the Neumann Problem in a polygonal domain ofR ², in which singular boundary data are imposed. Work performed under the auspices of G. N. A. F. A. of the Italian C. N. R.     

Finite element computation of absorbing boundary conditions for time-harmonic wave problems

This paper proposes a new method, in the frequency domain, to define absorbing boundary conditions for general two-dimensional problems. The main feature of the method is that it can obtain boundary conditions from the discretized equations without much knowledge of the analytical behavior of the solutions and is thus very general. It is based on the computation of waves in periodic structures and needs the dynamic stiffness matrix of only one period in the medium which can be obtained by standard finite element software. Boundary conditions at various orders of accuracy can be obtained in a simple way. This is then applied to study some examples for which analytical or numerical results are available. Good agreements between the present results and analytical solutions allow to check the efficiency and the accuracy of the proposed method.     

Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques

We consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary. As a paradigm for such problems, we simply refer to Stokes flow. After a brief review of the problem and of its well posedness, we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints, we exploit a penalty method to account for them. We perform the analysis of the method in terms of consistency, boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization. Since the additional terms introduced to account for the defective boundary conditions are non-local, we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy. In contrast to alternative discretization methods based on Lagrange multipliers accounting for the constraints on the boundary, the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation. Indeed, it does not require neither multipliers nor sub-iterations or additional adjoint problems with respect to the reference problem at hand.     

How to save a bad element with weak boundary conditions

The P₁-P₁ finite element pair is known to allow the existence of spurious pressure (surface elevation) modes for the shallow water equations and to be unstable for mixed formulations. We show that this behavior is strongly influenced by the strong or the weak enforcement of the impermeability boundary conditions. A numerical analysis of the Stommel model is performed for both P₁-P₁ and -P₁ mixed formulations. Steady and transient test cases are considered. We observe that the P₁-P₁ element exhibits stable discrete solutions with weak boundary conditions or with fully unstructured meshes.     

Quadratic approximation of solutions for boundary value problems with nonlocal boundary conditions

In this paper, using the quasilinearization method coupled with the method of upper and lower solutions, we study a class of second-order nonlinear boundary value problems with nonlocal boundary conditions. We establish some sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result.     

Optimal control for quasi-Newtonian flows with defective boundary conditions

We consider a generalized-Newtonian fluid with defective boundary conditions where only flow rates or mean pressures are prescribed on parts of boundary. The defect boundary condition problem is formulated as an optimal control problem in which a Neumann or Dirichlet boundary control is used for matching given flow rates or mean pressures. For the constrained optimization problem an optimality system is derived from which a solution of the problem is obtained. Computational algorithms are discussed and numerical results are also presented.     

Finite element solution of viscous jet flows with surface tension

A finite element method is used to study the effect of Reynolds number and surface tension on the expansion and contraction of jets of Newtonian liquids. For values of Reynolds numbers (based on tube diameter), below 14 the jets expand, and when Re > 14 the jets contract. For higher Reynolds numbers the jet diameter approaches a limiting value. It is also found that the surface tension has a considerable effect on low Reynolds number jet flows, becoming negligible at higher Reynolds numbers. As an example, if the surface tension parameter $\text{σ}\text{η}\text{u}$ is equal to unity, the creeping flow jet expansion is reduced by 4% relative to the case with no surface tension but when Re is equal to 20 and 50 the final jet diameters increase by only 0.2%. The calculated jet shapes are compared with available experimental results.     

Finite element approximation of some indefinite elliptic problems

We consider the finite element approximation of some indefinite Neumann problems in a domain of IR^N. From the Fredholm Alternative this kind of problem admits a solution if and only if the right hand term has zero mean value with respect to a measure whose density m is the solution of a homogeneous adjoint problem. The first step consists in the construction of piecewise linear finite element approximations m_h of m, showing their optimal rate of convergence both in energy and L^p norms. The functions m_h are then shown to be crucial in testing admissible data for the Neumann problem and also in its numerical resolution (actually, the standard Galerkin approximation may not be solvable without suitable perturbations of the data).     

Finite element approximation of a nonlinear diffusion problem

This paper deals with the finite element approximation of the nonlinear diffusion problem: −div (\vbgrad u\vb^p−2 grad u) = f. Glowinski and Marrocco[3] have been shown that the rate of convergence decreases as p increases. In this paper we show that the rate of convergence is optimal and independent of p. This theoretical result agrees with the numerical experiments reported in the last section.     

Molecular dynamics simulations of oscillatory Couette flows with slip boundary conditions

The effect of interfacial slip on steady-state and time-periodic flows of monatomic liquids is investigated using non-equilibrium molecular dynamics simulations. The fluid phase is confined between atomically smooth rigid walls, and the fluid flows are induced by moving one of the walls. In steady shear flows, the slip length increases almost linearly with shear rate. We found that the velocity profiles in oscillatory flows are well described by the Stokes flow solution with the slip length that depends on the local shear rate. Interestingly, the rate dependence of the slip length obtained in steady shear flows is recovered when the slip length in oscillatory flows is plotted as a function of the local shear rate magnitude. For both types of flows, the friction coefficient at the liquid–solid interface correlates well with the structure of the first fluid layer near the solid wall.     

Finite element solution for advection and natural convection flows

A new equal order velocity—pressure finite element procedure is presented for the calculation of 2-D viscous, incompressible flows of a recirculating nature. As in the finite difference procedures, velocity and pressure e uncoupled and the equations are solved one after the other. In this splitting-up method, an auxilary velocity field is computed first, which accounts for all contributions to the acceleration, except pressure, and satisfies the velocity boundary conditions. Then, the final velocities are evaluated by adding to the auxilary velocities pressure contributions which are computed to satisfy the continuity equation. The effectiveness is illustrated via example problems of 2-D advection and natural convection flows.     

Nonconforming finite element approximation of the Giesekus model for polymer flows

We present a numerical approximation of the Giesekus equation which is considered as a realistic model for polymer flows. We use nonconforming finite elements on quadrilateral grids which necessitate the addition of two stabilization terms. An appropriate upwind scheme is employed for the convective term. The underlying discrete Stokes problem is then analysed. Finally, numerical tests are presented in order to validate the code, illustrating its good behavior for large Weissenberg numbers. Comparisons with Polyflow® and with the literature are also carried out.     

Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.     

A parallel multipolar boundary element method for internal stokes flows

A multipolar expansion technique is applied to the Boundary Element Method (direct and indirect formulation) in order to solve the two-dimensional internal Stokes Flow with first kind boundary conditions. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the inner field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. In comparison with the direct BEM formulation, the indirect formulation is more stable with iterative solvers, and does not need to be preconditioned to obtain a fast rate of convergence. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvement. A good result in memory saving and computing time is obtained that enable us to run huge examples which are prohibitive for traditional BEM implementations.     

A solution technique for cathodic protection with dynamic boundary conditions by the boundary element method

This article is concerned with the application of the Boundary Element Method to cathodic protection problems of submerged structures using polarization curves depending upon time and formation potential. These curves have been adjusted from potentiostatic data obtained from in-situ experiments, yielding a nonlinear functional representation. The solution technique adopts stepwise linearized polarization curves and is employed for sufficiently small time steps. The influence of varying formation potential is introduced into the analysis under two alternative hypotheses here designated fictitious time and fictitious potential.     

Effect of inlet and outlet boundary conditions on swirling flows

Numerical simulations are conducted for both three-dimensional, turbulent flow in a multi-channel swirler and axisymmetric, isothermal, turbulent flow in combustion chambers using the standard κ−ε turbulence model. Calculations are first carried out for three-dimensional, isothermal and turbulent flow inside the swirler channels in order to derive the velocity profiles of both air and gas at the swirler outlets, which are used as inlet boundary conditions of the model combustor and can also be used in future studies for different combustors with the same type of swirler. In order to study the sensitivity of swirling flow inside the chamber to the inlet and outlet boundary conditions, different inlet velocity profiles and outlet boundary conditions are also employed. The results show that in the cases considered, the flow behaviour in the chamber is not very sensitive to the actual shape of the inlet velocity profiles provided the averages of the inlet axial, radial and azimuthal velocity components are separately preserved. Other conditions being equal, we find that the swirling flow performance in the combustor depends not only on the inlet swirl number, but also strongly on the relative magnitude of the radial velocity component at inlet and introduce a new dimensionless number N_r, analogous to the swirl number, to measure the relative importance of this quantity. Outlet boundary conditions have some influence near the outlet, but nearly no effect further upstream for the cases investigated.     

Finite element approximation of elliptic partial differential equations on implicit surfaces

The aim of this paper is to investigate finite element methods for the solution of elliptic partial differential equations on implicitly defined surfaces. The problem of solving such equations without triangulating surfaces is of increasing importance in various applications, and their discretization has recently been investigated in the framework of finite difference methods. For the two most frequently used implicit representations of surfaces, namely level set methods and phase-field methods, we discuss the construction of finite element schemes, the solution of the arising discretized problems, and provide error estimates. The convergence properties of the finite element methods are illustrated by computations for several test problems.     

Finite element approximation for the wave-particle interaction in weakly turbulent plasmas

A finite element method is presented for the numerical treatment of two-dimensional quasilinear equations. The method is applied to the electron-beam-plasma interaction. Strong influence of the initial fluctuation level on the evolution of the turbulence has been found.